Results 1  10
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22
New Results on Shortest Paths in Three Dimensions
 Proc. 20th Annual ACM Symposium on Computational Geometry
, 2004
"... We revisit the problem of computing shortest obstacleavoiding paths among obstacles in three dimensions. We prove new hardness results, showing, e.g., that computing Euclidean shortest paths among sets of “stacked ” axisaligned rectangles is NPcomplete, and that computing L1shortest paths among ..."
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Cited by 21 (0 self)
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We revisit the problem of computing shortest obstacleavoiding paths among obstacles in three dimensions. We prove new hardness results, showing, e.g., that computing Euclidean shortest paths among sets of “stacked ” axisaligned rectangles is NPcomplete, and that computing L1shortest paths among disjoint balls is NPcomplete. On the positive side, we present an efficient algorithm for computing an L1shortest path between two given points that lies on or above a given polyhedral terrain. We also give polynomialtime algorithms for some versions of stacked polygonal obstacles that are “terrainlike ” and analyze the complexity of shortest path maps in the presence of parallel halfplane “walls.”
Largest and Smallest Convex Hulls for Imprecise Points
 ALGORITHMICA
, 2008
"... Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we d ..."
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Cited by 10 (4 self)
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Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(n log n) to O(n^13), and prove NPhardness for some other variants.
Touring Convex Bodies  A Conic Programming Solution
 In Proceedings of the 17th Canadian Conference on Computational Geometry
, 2005
"... We study the problem of finding a shortest tour visiting a given sequence of convex bodies in R d. To our knowledge, this is the first attempt to attack the problem in its full generality: we investigate highdimensional cases (d ≥ 2); we consider convex bodies bounded by (hyper)planes and/or (hyper ..."
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Cited by 7 (1 self)
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We study the problem of finding a shortest tour visiting a given sequence of convex bodies in R d. To our knowledge, this is the first attempt to attack the problem in its full generality: we investigate highdimensional cases (d ≥ 2); we consider convex bodies bounded by (hyper)planes and/or (hyper)spheres; we do not restrict the start and the goal positions of the tour to be single points, we measure the length of the tour according to either Euclidean or L1 metric. Formulating the problem as a second order cone program (SOCP) makes it possible to incorporate distance constraints, which cannot be handled by a purely geometric algorithm. We implemented the SOCP in MATLAB and obtained its solution with the SeDuMi package. We ran computational experiments, which suggest that the proposed solution is practical. Finally, we present NPhardness results, showing that the assumptions we make in the statement of our problems are crucial for the problems to be tractable. 1
Competitive Online Approximation of the Optimal Search Ratio
 In Proc. 12th Annu. European Sympos. Algorithms, volume 3221 of Lecture Notes Comput. Sci
, 2004
"... How e#ciently can we search an unknown environment for a goal in unknown position? How much would it help if the environment were known? We answer these questions for simple polygons and for general graphs, by providing online search strategies that are as good as the best o#ine search algorithm ..."
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Cited by 6 (3 self)
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How e#ciently can we search an unknown environment for a goal in unknown position? How much would it help if the environment were known? We answer these questions for simple polygons and for general graphs, by providing online search strategies that are as good as the best o#ine search algorithms, up to a constant factor. For other settings we prove that no such online algorithms exist.
Watchman Route in a Simple Polygon with a Rubberband Algorithm
"... So far, the best result in running time for solving the fixed watchman route problem (i.e., shortest path for viewing any point in a simple polygon with given start point) is O(n 3 log n), published in 2003 by M. Dror, A. Efrat, A. Lubiw, and J. Mitchell. – This paper provides an algorithm with κ(ε) ..."
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Cited by 5 (5 self)
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So far, the best result in running time for solving the fixed watchman route problem (i.e., shortest path for viewing any point in a simple polygon with given start point) is O(n 3 log n), published in 2003 by M. Dror, A. Efrat, A. Lubiw, and J. Mitchell. – This paper provides an algorithm with κ(ε) · O(kn) runtime, where n is the number of vertices of the given simple polygon Π, and k the number of essential cuts; κ(ε) defines the numerical accuracy in dependency of a selected constant ε> 0. Moreover, our algorithm is significantly simpler, easier to understand and implement than previous ones for solving the fixed watchman route problem. 1
Approximate Shortest Path Algorithms for Sequences of Pairwise Disjoint Simple Polygons
"... Assume that two points p and q are given and a finite ordered set of simple polygons, all in the same plane; the basic version of a touringasequenceofpolygons problem (TPP) is to find a shortest path such that it starts at p, then visits these polygons in the given order, and ends at q. This pap ..."
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Cited by 4 (4 self)
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Assume that two points p and q are given and a finite ordered set of simple polygons, all in the same plane; the basic version of a touringasequenceofpolygons problem (TPP) is to find a shortest path such that it starts at p, then visits these polygons in the given order, and ends at q. This paper describes four approximation algorithms for unconstrained versions of problems defined by touring an ordered set of polygons. It contributes to an approximate and partial answer to the previously open problem “What is the complexity of the touringpolygons problem for pairwise disjoint, simple and not necessarily convex polygons? ” by providing κ(ε)O(n) approximation algorithms for solving this problem, either for given start and end points p and q, or with allowing to have those variable, where n is the total number of vertices of the given k simple and pairwise disjoint polygons; κ(ε) defines the numerical accuracy in dependency of a selected ε> 0. 1 Contributions of this Paper According to [1], “one of the most intriguing open problems” identified by their results “is to determine the complexity of the fixed TPP for pairwise disjoint nonconvex simple polygons”. In this paper, we focus on the unconstrained fixed TPP (i.e., given start and end point of the path) and floating TPP (i.e., no given start or end point) under the condition that the convex hulls of the input polygons Pi are pairwise disjoint, but the polygons Pi itself may be nonconvex. Algorithm 2 in Section 2 partially answers the stated open problem for the fixed TPP by providing an approximation algorithm running in time κ(ε) · O(n), where n is the total number of vertices of all polygons. The solution technique proposed in [1] can only handle the fixed TPP, the fixed safari problem, and the fixed watchman route problem, all for convex polygons only. Our solution technique is suitable for solving both the fixed and the floating TPP with the same time complexity,
Existence of simple tours of imprecise points
 IN PROC. 23RD ANNU. EUROPEAN WORKSHOP ON COMPUTATIONAL GEOMETRY
, 2007
"... Assume that an ordered set of imprecise points is given, where each point is specified by a region in which the point may lie. This set determines an imprecise polygon. We show that it is NPcomplete to decide whether it is possible to place the points inside their regions in such a way that the res ..."
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Cited by 3 (0 self)
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Assume that an ordered set of imprecise points is given, where each point is specified by a region in which the point may lie. This set determines an imprecise polygon. We show that it is NPcomplete to decide whether it is possible to place the points inside their regions in such a way that the resulting polygon is simple. Furthermore, it is NPhard to minimize the length of a simple tour visiting the regions in order, when the connections between consecutive regions do not need to be straight line segments.
Shortest Paths with SinglePoint Visibility Constraints
"... This paper studies the problem of finding the shortest path between two points in presence of singlepoint visibility constraints. In this type of constraints, there should be at least one point on the output path from which a fixed viewpoint is visible. The problem is studied in various domains ..."
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Cited by 3 (2 self)
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This paper studies the problem of finding the shortest path between two points in presence of singlepoint visibility constraints. In this type of constraints, there should be at least one point on the output path from which a fixed viewpoint is visible. The problem is studied in various domains including simple polygons, polygonal domains, polyhedral surfaces. The method is based on partitioning the boundary of the visibility region to a number of intervals according to shortest path structure of their points to both source and destination. The result for the two dimensional domains is worstcase optimal
Visiting a Sequence of Points with a BevelTip Needle ⋆
"... Abstract. Many surgical procedures could benefit from guiding a beveltip needle along circular arcs to multiple treatment points in a patient. At each treatment point, the needle can inject a radioactive pellet into a cancerous region or extract a tissue sample. Our main result is an algorithm to s ..."
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Cited by 1 (0 self)
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Abstract. Many surgical procedures could benefit from guiding a beveltip needle along circular arcs to multiple treatment points in a patient. At each treatment point, the needle can inject a radioactive pellet into a cancerous region or extract a tissue sample. Our main result is an algorithm to steer a beveltip needle through a sequence of treatment points in the plane while minimizing the number of times that the needle must be reoriented. This algorithm is related to [6] and takes quadratic time when consecutive points in the sequence are sufficiently separated. We can also guide a needle through an arbitrary sequence of points in the plane by accounting for a lack of optimal substructure.